[FOM] FW: FOM Real Numbers
Matt Insall
montez at fidnet.com
Fri May 9 09:58:36 EDT 2003
...
>Insall:
>
>Now, I shall try my hand at explaining what I think can clear this up.
>My explanation will involve the use of urelements, which are objects
>in a set theory that are not themselves sets.
Makarov:
We do not need urelements to explain what real numbers are.
Insall:
It is not a question of ``need'', when dealing with others who can ``see''
the field of real numbers. Thus you are correct, but why do you bring this
up?
There are many things we do when we do not ``need'' to do them.
Makarov:
In mathematics, every mathematical concept Z exists only in the context of a
certain mathematical theory T (let us call this theory T as the host theory
of the concept Z).
Insall:
I would not limit this to mathematics. Similar contextualization principles
apply in all of science and life.
Makarov:
For the most of mathematicians (not set-theorists) the
host theory for the concept "real number" is the theory of the field R.
Insall:
Well, how do you respond to the person who knows not what the field R is?
Do you provide axioms for fields and then stop and say ``Now do you know
what I mean?'' I guess not. Do you include the order-theoretic axioms
and then say ``Now do you know what the real numbers are?'' For some
audiences, you might, since enough such axioms identify the real number
system up to isomorphism. But how do you respond to someone who asks
what are the elements of the field R? If you begin to construct N, and then
Z, and then Q, and then R, all from the empty set, then you have not, as
I have been learning, the agreement of certain analysts (R-scientists) who
do not consider the field R to BE the set of equivalence classes of Cauchy
sequences of rational numbers. The set of equivalence classes of Cauchy
sequences of rationals is merely a MODEL of the set of axioms that
R-scientists
have written down that are satisfied by THE (one and only) field R.
Makarov:
For
set-theorists the host theory is a set theory (ZFC, ZFA, NF, ... ). So
asking. what real numbers are we should add "in what theory". And for the
most of mathematicians the answer is well-known:
real numbers are elements of the field R.
Insall:
I am responding, however, to the question as it is asked, not necessarily
as it SHOULD BE asked. In the message to which I was replying, it seemed
to me that Heck was discussing the question ``What are the real numbers?'',
not ``What are the real numbers in ZFC?''. My response is that the real
numbers
are what I see when I do real analysis, and I can model that structure in
various ways, using ZF, using ZFC, using ZFA, using NF, etc. But the most
realistic
model, in my opinion, is in ZFA (ZF with urelements), as I described it,
with
an added predicate to select the real numbers from among the urelements, and
with added axioms that specify (enough of) the properties that the set R of
real numbers enjoys.
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